I have a very basic question regarding algebraic model theory. I am trying to read *Espaces de Berkovich, polytopes, squelettes et théorie des modèles* (MSN) by Antoine Ducros. The relevant section is Section 0.31. Let me give the setup: Let $K$ be a valued field and let $\mathscr{C}_K$ be the category of non-trivially valued, algebraically closed field extensions of $K$. We write $\mathscr{L}_\mathrm{val}$ for the language of valued fields, having three sorts (the valued field, its residue field and the value group).

Let $\mathscr{S}$ be a product of sorts. For each model of the theory of non-trivially valued, algebraically closed field extensions of $K$, i.e. for each $F \in \mathscr{C}_K$, we can then make sense of $\mathscr{S}(F)$, and this gives a functor $\mathscr{S} \colon \mathscr{C}_K \to \mathrm{Set}$. If $\Phi(x)$ is a formula of $\mathscr{L}_\mathrm{val}$ (with parameters in $K$), whose free variables $x$ live in $\mathscr{S}$, then the subfunctor of $\mathscr{S}$ mapping $F$ to $\{x \in \mathscr{S}(F) \mid \Phi(x)\}$ is called a *definable subfunctor* of $\mathscr{S}$.

In general, an abstract functor from $\mathscr{C}_K$ to $\mathrm{Set}$ is called *definable* if it is isomorphic to a definable subfunctor of some product of sorts.

Now it is claimed that a scheme of finite type $\mathscr{X}$ over $K$ gives rise to a definable functor on $\mathscr{C}_K$. I don't understand why this is true. If $\mathscr{X}$ is affine, then I believe that the associated functor should simply be the “functor of points”. Namely, if $\mathscr{X}$ is isomorphic to $\mathrm{Spec}(K[T_1, \dots, T_n] / \langle f_1, \dots, f_r\rangle)$, then its associated functor is isomorphic to the definable subfunctor of $F \mapsto F^n$, given by the equations $f_i$. I have no idea, what is meant in the non-affine case. I would be very glad about some clarification. Also, if some of my writing above hints at some misunderstanding on my side, please let me know.